Fundamentals of Plasma Physics

5.2 The Landau problem 155

a relationship betweenn 1 andφ 1 .Another relationship betweenn 1 andφ 1 is Poisson’s
equation
∂^2 φ 1
∂x^2

=−

n 1 q

ε 0

. (5.26)

Replacing∂/∂xbyik,Eq.(5.26) becomes

k^2 φ 1 =

n 1 q

ε 0

. (5.27)

Combining Eqs.(5.25) and (5.27) gives the dispersion relation

1+

q^2

k^2 mε 0

∫∞

−∞

k

(ω−kv)

∂f 0

∂v

dv=0. (5.28)

This can be written more elegantly by substituting forf 0 using Eq.(5.20), defining the non-
dimensional particle velocityξ =v/vT, and the non-dimensional phase velocityα=
ω/kvT to give

1 −

1

2 k^2 λ^2 D

1

π^1 /^2

∫∞

−∞

dξ

1

(ξ−α)

∂

∂ξ

e−ξ

2

=0. (5.29)

or
1+χ=0 (5.30)
where the electron susceptibility is

χ=−

1

2 k^2 λ^2 D

1

π^1 /^2

∫∞

−∞

dξ

1

(ξ−α)

∂

∂ξ

e−ξ

2

. (5.31)

In contrast to the earlier two-fluid wave analysis where in effect the zeroth, first, and second
moments of the Vlasov equation were combined (continuity equation, equation of motion,
and equation of state), here only the Vlasov equation is involved. Thus theVlasov equa-
tion contains all the information of the moment equations and more. The Vlasov method
therefore seems a simpler and more direct way for calculating the susceptibilities than the
fluid method, except for a serious difficulty: the integral in Eq.(5.31) is mathematically
ill-defined because the denominator vanishes whenξ=α(i.e., whenω =kvT). Be-
cause it is not clear how to deal with this singularity, theζintegral cannot be evaluated and
the Fourier method fails. This is essentially the same as the problem encountered influid
analysis whenω/kbecame comparable to

√

κT/m.

5.2.2 Landau method: Laplace transforms

Landau (1946) argued that the Fourier problem as presented above is ill-posed and showed
that the linearized Vlasov-Poisson problem should be treated as aninitial valueproblem,
rather than as a normal mode problem. The initial value point of view is conceptually re-
lated to the analysis of single particle motion in sawtooth or sine waves.Before presenting
the Landau analysis of the linearized Vlasov-Poisson problem, certain important features
of Laplace transforms will now be reviewed.
The Laplace transform of a functionψ(t)is defined as

ψ ̃(p)=

∫∞

0

ψ(t)e−ptdt (5.32)